Lecture Today: Chapter 14 1) Archimedes' Principle 2 .

PES 2130 Fall 2014, SpendierLecture 29/Page 1Lecture today: Chapter 141) Archimedes' Principle2) Ideal Fluids in Motion3) Bernoulli's EquationAnnouncements:- Exam 3 graded (current class grade written in top right corner of exam 3)- HW11 was due Wednesday extension until Nov. Friday12pm (drop off at myoffice on or before noon this Friday)- Wednesday lecture review for FINAL (Dec 17th 5:20-7:20 )Last time:Fluid - Anything that can flow either a liquid or a gaspressure force per unit areaF N unit 2 P Pa.Pascal (SI unit for pressure)p A m true for uniform force over a flat areaFluid pressure acts perpendicular to any surface in the fluid.density mass per unit volumem kg unit 3 V m Expression for hydrostatic pressure as a function of depth or altitude.p p0 hgp0 the pressure due to 1 atmosphere(only applicable to constant density fluids!)Pascal's Principle"A change in the pressure applied to an enclosed incompressible fluid is transmittedundiminished to every portion of the fluid and to the walls of its container".

PES 2130 Fall 2014, SpendierLecture 29/Page 2Buoyancy DEMO Why Does Coke Sink & Diet Coke Float in Water?Immerse sealed cans of Coke and Diet Coke in a tank of water. The can of Cokeimmediately sinks, while the can of Diet Coke floats.BuoyancyWhether an object sinks or floats depends on its buoyancy. An object placed in waterexerts a downward force on the water. The water, however, pushes back. Archimedes’principle states that the buoyant force exerted by water or any other fluid on an object isequal to the weight of the water displaced by the object. If the weight of displaced waterexceeds the weight of the object, the object floats. Otherwise, it sinks. This, in part,explains why metal ships float. It also explains why the can of Diet Coke sinks. Becausethe two cans exhibit identical shapes and sizes, they displace equal amounts of waterwhen submerged. But the fact that the can of Coke sinks means it must weigh more thanthe amount of water it displaces, whereas the can of Diet Coke weighs less.Mathematical ExplanationBoth the Diet Coke and Coke cans contain 12 fluid oz., or 355 milliliters, of liquid. Bothbeverages consist primarily of water. The primary difference lies in the sweetener. Cokecontains about 325 mL of water, with a density of 1.0 g/mL, and 39 g of sugar. Thecontents of the can therefore weigh 325 g 39 g 364 g. The can of Diet Coke,however, contains only 0.3 g of aspartame. It therefore consists almost entirely of waterand the can’s contents therefore weigh about 355 g. This difference in weight makes thecan of Diet Coke sufficiently buoyant to float. In terms of density, the density of DietCoke is roughly 1.00 g/mL, the same as water. The Coke, however, exhibits a density of1.03 g/mL.1) Archimedes' Principle (Buoyancy)The magnitude of the buoyant force B always equals the weight of the fluid displaced bythe object. This statement is known as Archimedes’s principle:B mf gmf mass of fluid that is displaced by bodyWhen a body is fully or partially submerged in a fluid, a buoyant force B from thesurrounding fluid acts on the body. The force is directed upward and has amagnitude equal to the weight mf g of the fluid that has been displaced by the body.Fb m f gmf mass of fluid that is displaced by body

PES 2130 Fall 2014, SpendierLecture 29/Page 3DEMO: Cartesian DiverSqueeze the bottle hard enough; you put pressure on the diver. That causes the airbubble to get smaller and the entire diver to become MORE DENSE than the wateraround it and the diver sinks. When you release the pressure, the bubble expands,making the diver less dense (and more buoyant) and, alas, it floats back up.Example:An iceberg floating in seawater, is extremely dangerous because most of the ice is belowthe surface. This hidden ice can damage a ship that is still a considerable distance fromthe visible ice. What fraction of the iceberg lies below the water level?Density of seawater, ρsw 1030 kg/m3Density of ice, ρi 917 kg/m3When an object floats, the net force on it will be zero.Volume of submerged object displaces an amount of liquid whose weight is equal to theweight of the object.Aside: Ships float because they displace more water than it weights.

PES 2130 Fall 2014, SpendierLecture 29/Page 42) Ideal Fluids (Fluid flow, Continuity Equation)Thus far, our study of fluids has been restricted to fluids at rest. We now turn ourattention to fluids in motion. The motion of fluids is a very complex phenomena. Tosimplify as much as possible, we usually make three assumptions:1. The fluid is incompressible as it flows. (Gases are hard to compress once they aremoving, so it’s a pretty good assumption for them too.) The density of an incompressiblefluid is constant.2. The flow is steady. In steady flow, the velocity of the fluid at each point remainsconstant. Steady flow is also called laminar. Non-steady is called turbulent.3. The fluid is nonviscous. Viscosity is analogous to friction.In a nonviscous fluid, internal friction is neglected. An object moving through the fluidexperiences no viscous force, hence fluid flow is steady.(Demo of corn starch non-Newtonian fluid)Corn starch demo:Add ¼ cup of dry cornstarch to the bowl. Add about 1/8 cup (2 tablespoons, or 30 cm3)of water to the corn starch and stir slowly. Add water slowly to the mixture, with stirring,until all of the powder is wet.Continue to add water until the cornstarch acts like a liquid when you stir it slowly. Whenyou tap on the liquid with your finger, it shouldn't splash, but rather will become hard. Ifyour mixture is too liquid, add more cornstarch. Your goal is to create a mixture that feelslike a stiff liquid when you stir it slowly, but feels like a solid when you tap on it withyour finger or a spoon.Why does the cornstarch mixture behave like this?Think of a busy sidewalk. The easiest way to get through a crowd of people is to moveslowly and find a path between people. If you just took a running start and headedstraight for the crowd of people, you would quickly slam into someone and you wouldn'tget very far. This is similar to what happens in the cornstarch mixture. The solidcornstarch acts like a crowd of people. Pressing your finger slowly into the mixtureallows the cornstarch to move out of the way, but tapping the mixture quickly doesn'tallow the solid cornstarch particles to slide past each other and out of the way of yourfinger.We use the term “viscosity” to describe the resistance of a liquid to flow.Water has very low viscosity!

PES 2130 Fall 2014, SpendierLecture 29/Page 5Equation for the flow of an ideal fluid:Now let's investigate the assumption that fluids are incompressible a bit further. You mayhave noticed that you can increase the speed of the water emerging from a garden hose bypartially closing the hose opening with your thumb. Apparently the speed v of the waterdepends on the cross-sectional area A through which the water flows.Since we assume fluids are incompressible, as they flow an equal volume must bemoving into and out of each part of the fluid’s container. Consider the tube below withtwo cylinders of fluid that have the same volume.Setting the volumes equal V1 V2A1 x1 A2 x2and dividing by time x xA1 1 A2 2 t1 t2gives relation between speed and cross-sectional area which is called the ContinuityEquation for the flow of an ideal fluid:A1v1 A2 v2It tells us that the flow speed increases when we decrease the cross-sectional areathrough which the fluid flows.

PES 2130 Fall 2014, SpendierLecture 29/Page 63) Bernoulli's EquationNow let’s add elevation change to the tube, i.e. a tube that connects a lower point and ahigher point in space. As a fluid moves through a region where its speed and/or elevationabove the Earth’s surface changes, the pressure in the fluid varies with these changes.The relationship between fluid speed, pressure, and elevation was first derived in 1738 bythe Swiss physicist Daniel Bernoulli.Consider the case of water flowing through a smooth pipe.The Bernoulli Equation is derived from conservation of energy and work-energy ideasthat come from Newton's Laws of Motion. (Look in book for derivation/proof)Let y1, v1, and p1 be the elevation, speed, and pressure of the fluid entering at the left, andy2, v2, and p2 be the corresponding quantities for the fluid emerging at the right. Byapplying the principle of conservation of energy to the fluid, can show that thesequantities are related byorBernoulli’s equation is strictly valid only to the extent that the fluid is ideal. If viscousforces are present, thermal energy will be involved.Check: Let us apply Bernoulli’s equation to fluids at rest, by putting v1 v2 0Which we derived already in Lecture 27!

PES 2130 Fall 2014, SpendierLecture 29/Page 7Example:Water enters a house through a pipe with an inside diameter of 2.0 cm at an absolutepressure of 4.0 x 105 Pa (about 4 atm). A 1.0 cm diameter pipe leads to the second-floorbathroom 5.0 m above. When the flow speed at the inlet pipe is 1.5 m/s, findi) the flow speedii) and pressure,in the bathroom.DEMO: cornstarch in water grows weird shapes due to vibrations of a speakerNon-Newtonian Fluid (cornstarch and water). The cornstarch acts almost like a solidwhen its impacted quickly fast vibrations of a speaker cone makes the fluid squirmaround.https://www.youtube.com/watch?v SYMvOxIsES4

the Swiss physicist Daniel Bernoulli. Consider the case of water flowing through a smooth pipe. The Bernoulli Equation is derived from conservation of energy and work-energy ideas that come from Newton's Laws of Motion. (Look in book for derivation/proof) Let y 1, v 1, and p 1 be the elevation, speed, and pressure of the fluid entering at the .